又化简了一下,最终结果是$ - \left( {\left( { - 96abc - 32a{c^2} - 32b{c^2} + 32{c^3} - 160abd - 224acd - 224bcd - 32{c^2}d - 128a{d^2} - 128b{d^2} - 128c{d^2} + 128{d^3} + 8ac\left| {a + b - 2c - \left| {a - b} \right|} \right| + 8bc\left| {a + b - 2c - \left| {a - b} \right|} \right| - 16{c^2}\left| {a + b - 2c - \left| {a - b} \right|} \right| + 24ad\left| {a + b - 2c - \left| {a - b} \right|} \right| + 24bd\left| {a + b - 2c - \left| {a - b} \right|} \right| + 16cd\left| {a + b - 2c - \left| {a - b} \right|} \right| + 8c\left| {a - b} \right|\left| {a + b - 2c - \left| {a - b} \right|} \right| + 24d\left| {a - b} \right|\left| {a + b - 2c - \left| {a - b} \right|} \right| - 8ac\left| {a + b - 2c + \left| {a - b} \right|} \right| - 8bc\left| {a + b - 2c + \left| {a - b} \right|} \right| + 16{c^2}\left| {a + b - 2c + \left| {a - b} \right|} \right| - 24ad\left| {a + b - 2c + \left| {a - b} \right|} \right| - 24bd\left| {a + b - 2c + \left| {a - b} \right|} \right| - 16cd\left| {a + b - 2c + \left| {a - b} \right|} \right| + 8c\left| {a - b} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| + 24d\left| {a - b} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| - 8c\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| + } \right.} \right. - 8{c^2}\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - 32{d^2}\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 16ac\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 16bc\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 16cd\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 8c\left| {a - b} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 4c\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 8d\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| + 8ab\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 24ad\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 24bd\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 8d\left| {a - b} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2a\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2b\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 8d\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2\left| {a - b} \right|\left| {a + b - 2c - \left| {a - b} \right|} \right|8{c^2}\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 32{d^2}\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 4c\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - 16ac\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 16bc\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 16cd\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 8c\left| {a - b} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2a\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2b\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - 2\left| {a - b} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + \left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - 8ab\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 24ad\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 24bd\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 8d\left| {a - b} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2a\left| {a + b - 2c - \left| {a - b} \right|} \right|2a\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2b\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 4c\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 8d\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 2\left| {a - b} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2b\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 4c\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2\left| {a - b} \right|\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 2\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {\left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a - b} \right| + a + b + 2c - 4d} \right| + 4a\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 4b\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 8d\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 2\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| \\± \left. {\left. {2\sqrt {\frac{1}{4}{{(\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c - \left| {a - b} \right|} \right| + \left| {a - b} \right| - a - b - 2c - 4d)}^2}{{(\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - \left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c - \left| {a - b} \right|} \right| - \left| {a + b - 2c + \left| {a - b} \right|} \right| + 6a + 6b + 4c)}^2}{{(\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a - b} \right| + a + b + 2c + 4d)}^2} - 4096abcd( - \left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right| - \left| {a - b} \right| + a + b + 2c + 4d)(\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a - b} \right| + a + b + 2c + 4d)} } \right)/16( - \left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right| - \left| {a - b} \right| + a + b + 2c + 4d)(\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a - b} \right| + a + b + 2c + 4d)} \right)$我验证了10组实数,未发现问题,只是a,b,c,d都不能为0,否则分母为0.而且分母也很麻烦,整理后发现当且仅当a,b,c,d都为正数或都为负数时,分母为正数,即根号前面带负号,不然,如1,2,-1,-9,就带正号,所以只能写±号.这引发一个问题:四元情形能否像1#"提示"中二元和三元的情形那样,给出一个表达式,对任何实数都行得通? |